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17 Jul 2017, 10:20
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If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, how many positive factors does \(n\) have?
(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\).
(2) \(n\) has only two prime factors.
(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\).
(2) \(n\) has only two prime factors.
01 Sep 2018, 09:42
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Official Solution:
If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, how many positive factors does \(n\) have?
(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\).
This statement implies that both \(x\) and \(y\) are primes (a prime number does not have a factor that is greater than 1 and less than itself; it has only two factors: 1 and itself). Now, if \(x\) and \(y\) are different primes, then the number of factors of \(n\) will be \((5+1)(7+1)=48\). However, if \(x=y\), then \(n = x^5*y^7=x^{12}\) and it will have 13 factors. Not sufficient.
(2) \(n\) has only two prime factors.
If those primes are \(x\) and \(y\), then the number of factors of \(n\) will be \((5+1)(7+1)=48\). But if, say, \(x=2*3=6\) and \(y=2\), then \(n = x^5*y^7=2^{12}*3^5\) and it will have \((12+1)(5+1)=78\) factors. Not sufficient.
(1)+(2) Since from (2) \(n\) has only two prime factors, then from (1) it follows that \(x\) and \(y\) are different primes; so the number of factors of \(n\) will be \((5+1)(7+1)=48\). Sufficient.
Answer: C
If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, how many positive factors does \(n\) have?
(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\).
This statement implies that both \(x\) and \(y\) are primes (a prime number does not have a factor that is greater than 1 and less than itself; it has only two factors: 1 and itself). Now, if \(x\) and \(y\) are different primes, then the number of factors of \(n\) will be \((5+1)(7+1)=48\). However, if \(x=y\), then \(n = x^5*y^7=x^{12}\) and it will have 13 factors. Not sufficient.
(2) \(n\) has only two prime factors.
If those primes are \(x\) and \(y\), then the number of factors of \(n\) will be \((5+1)(7+1)=48\). But if, say, \(x=2*3=6\) and \(y=2\), then \(n = x^5*y^7=2^{12}*3^5\) and it will have \((12+1)(5+1)=78\) factors. Not sufficient.
(1)+(2) Since from (2) \(n\) has only two prime factors, then from (1) it follows that \(x\) and \(y\) are different primes; so the number of factors of \(n\) will be \((5+1)(7+1)=48\). Sufficient.
Answer: C
24 Oct 2019, 04:15
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Hi Bunuel
In this question for explanation for statement 1: It says that " what if x=y ( i.e. both prime numbers are same) AND in another line written to prove that answer is C is that: x and y are different prime numbers( from statement 1). Isn't this contradicting, since you are assuming two differenet scenarios from statement 1 only.
Could you pl. explain.
In this question for explanation for statement 1: It says that " what if x=y ( i.e. both prime numbers are same) AND in another line written to prove that answer is C is that: x and y are different prime numbers( from statement 1). Isn't this contradicting, since you are assuming two differenet scenarios from statement 1 only.
Could you pl. explain.
